Bernoulli Number Puns

I have 2 functions. In one the user inputs n and it returns a list of the 1st n Bernoulli numbers, recursively generating the Kth number by doing some math on the (k-1)th number. The other function generates and returns only the Nth Bernoulli number. Both functions diverge wildly from what the outputs should be at around n=13 and I'm wondering if this is unavoidable due to rounding errors? I thought errors might be compounding in the 1st function which is why I wrote the second but I'm seeing similar behavior in both. I'm only including the 2nd function here for brevity. Thanks for any insight!

https://pastebin.pl/view/1f89b044

Can anyone explain Bernoulli numbers and / or link to any helpful material? Even stuff that I should know beforehand, etc. I'm attempting to learn them to work out how to use the tan function (for a Butterworth filter algorithm for music).

I've found this (link: https://medium.com/@starkcoffee/discovering-adas-bernoulli-numbers-part-1-65bdfeb7cc8f) for Bernoulli numbers, but I'm still having trouble and only get to about half-way down the page almost. I'm working everything out by intuitive leaps mainly but it's pretty slow. There also doesn't seem to be a lot of resources around

Thanks team

This is a version of a code I made in Python using mainly properties of Pascal's (with binomial coefficients) triangle to compute Bernoulli numbers (My only other algorithms use the positive recursive definition and generating functions) . -5.31870446941632e+1769 is ^(B)1000 in approx. 14 seconds of calculation . I am very unsure of its time complexity and know there are far more efficient ways to calculate them. I am new to number theory, which I am learning by myself for now.

I was wondering if anyone had experience with Bernoulli numbers here and if so would you be so kind as to tell where to start learning about these ? I read about their complex links to the Riemann zeta function, combinatorics, a variety of series and that they have a bunch of different representations and definitions...

I have a difficult time wrapping my head around all their implications and I would be interested to know which methods and definitions should I read about first and what mathematical background would they require ?

(I'm new here sorry if the format is not adequate or I am unclear, I will give any precision necessary, sorry this is not a vocabulary I am used to; not a native speaker)

https://preview.redd.it/cqzdftsr9d551.png?width=1493&format=png&auto=webp&s=32c72bdaab00ee2e18213a4061faadc8fcea84f2

Anyone know of any articles or books/chapters that explain Bernoulli Numbers and their application?

I'm having trouble finding the exponential generating function for Bernoulli numbers given that B0=0 and the sum from k=0 to n of ((n+1) choose k )*Bk = 0, for each natural number n where Bn denotes the nth Bernoulli number. Sorry for the horrible formatting but it should suffice. I've set f(x) to be the EGF with coefficients Bn and multiplied it with e^x in its power series form. After multiplying I divided the inner sum to the sum of k=0 to n-1 of n choose k * Bk(which should be equal to zero) and the last element which is just Bn. I end up with the relation f(x)e^(x)=f(x) which makes no sense. I'm missing a piece somewhere. All help is appreciated, thanks.

I would expect the first program to be a code that added 2 numbers, or something simple like that.

Could it be that this is the oldest surviving code from Ada Lovelace but that she wrote other simpler programs before it?

I wasn't able to find anything about this on math.SE or various google searches, but I might well have missed something.

A practical algorithm for making a uniform pick in [1, n] is to extend the range to the nearest power of two and then pick in that extended range, i.e. using ⌈log2 n⌉ bits. The result is the base-2 representation of a number, which is returned if it falls in the desired range; otherwise, we repeat the procedure. This is fine for practical computing purposes but it's not guaranteed to terminate.

The program ERF calculates the error function:

erf(x) = 2 / √π * ∫ e^-(t^2) dt from t = 0 to t = x

The calculation uses the series:

erf(x) = 2 * e^(-(x^2)) / √π * Σ( 2^n * x^(2n+1) / (1 * 3 * ... * (2n+1)) from n=0 to ∞

The program BERN calculates Bernoulli numbers for positive integers n > 2. The calculator is set to Radian mode.

The formula used:

B_n = (-cos(n * π/2))^((n+2)/2) * 2 * n! * Σ(2 * j * π)^(-n) from j=1 to ∞

Chebyshev Polynomials

The program CHEBY calculates the value of a first-order Chebyshev polynomial using the recurring formula where:

T_n(x) = 2 * x * T_n(x) - T_(n-1)(x)

where T_0(x) = 1, T_1(x) = x

Link: http://edspi31415.blogspot.com/2019/10/hp-42sdm42-error-function-bernoulli.html

The wiki page on them goes just barely over my head.

We are given the following recursion relation for the Bernoulli numbers:

[;B_j = \sum_{n=0}^{j-1} \frac{B_n}{n!(j-n+1)!};]

Use it to prove [;B_3 = B_5 =B_7 =\ldots=0;]

So my attempt was using induction, but I'm stuck in the final step for [;k=n+1;] (assuming I want to prove [;B_{2k+1}=0;] for every k). I'm utterly stuck, any help?

Thanks!!

Hi. I'm working on a personal project and need to make sure the work I'm doing is mathematically sound. For the questions below, is my approach correct, or is there a correct/better way of doing things?

I have conducted 20 Bernoulli trials. My observed outcome is 8 successes.

It was proposed that $p_{suc}$ was 0.2. As the number of successes follow a binomial distribution, I have calculated the standard deviation as $\sqrt(npq)$ = $\sqrt(200.20.8)$ = $\sqrt(3.2)$ = 1.79

The proposed mean was 4, so my observed value is therefore + 2.22 standard deviations from the mean, and so my p value is 0.0264. So this is significant by my measure (<0.05). However is it valid to say that there is a 97.4% chance that $p_{suc}$> 0.2?

Also, I want to define a range of values for $p_{suc}$ given my observed number of successes, which I can be 95% confident in. So my estimate is $8/20$ = 0.4 +/- $x$.

Is there a way to calculate $x$, or can I use brute force to simulate 20 trials say 10,000+ times for different $p_{suc}$ values, and use an algorithm to find the upper and lower cut-off values for $p_{suc}$ whereby my observed success count of 8 appears in < 5% of simulations?

Thanks for any help.

I think it is interesting that this holds as far as I've checked. If it isn't prime, that fraction isn't an integer.

n is offset assuming you consider the Bernoulli numbers starting at 0.

(Note: I asked this question before on the Stats Stackexchange, but did noy get any authoritative answer. I hope my luck is better with Reddit).

Suppose you have a series of n trials, where the probability of success in each trial is p. The distribution of the number of successful trials follows a Binomial distribution with parameters (n, p). The mean is given by np whereas the variance is np(1-p). So far so good: this is pretty mundane Stats 101 stuff.

But suppose now that I only knew about the number m of successful trials, and had no knowledge of the total number n of trials, which is the variable I am interested in estimating. For example, I knew I had 100 successful trials, where each trial had a 0.1 chance of success. Is there a known probability distribution that describes the probable outcomes for n, the total number of trials? Estimating the mean is easy: m/p. But what about variance and other measures?

What if each success had a different (but known) chance of success? Suppose I had the following records:

• success1 (with p=0.1)
• success2 (with p=0.1)
• success3 (with p=0.2)

Again, a good estimation of the total number of trials can be obtained by simply summing 1/p for each successful trial. In this case that number is 10+10+5=25. But what about variance and other measures?

So i was reading up on Ada Lovelace, who's credited with, as you probably know, creating the first every algorithm; to compute Bernoulli numbers.

So i tried to read in to them but i ran in too maths not with numbers, not even with letters, with greek looking shit.

So can any of you folk explain Bernoulli numbers to me in any way or is it so far in mathematician territory it'd take too long?

Using this generating function, what exactly do you do with the t in the summation? I don't understand how the bernoulli numbers are generated from this...?

Every thread that I found has no comments, and when I tried to make a thread in eli5, no one commented either.